This satisfies OP’s rule since the 3rd-place was Eugene Debs the Socialist with only 3.4%.
Cox’s V.P. running mate suffered a severe paralytic disease a year after the electoral failure, but would be more successful in his future election campaigns: he was eventually elected Governor of New York and then to four terms as President of the United States.
It’s only more complex than it seems if you go back before the Civil War when the current party system didn’t exist and no one cared about anything other than the Electoral vote. But that’s a bar bet nitpick, not a GQ.
If you want to be absolutely pedantic, George Washington was elected President in 1788 with 0 popular votes because no state chose their electors by popular vote until 1789 when the Constitution was ratified.
In 1820 James Monroe was elected President with all but one electoral votes. For some reason, DeWitt Clinton, who was also a Democrat-Republican, received 1,893 votes. One elector cast a vote for John Quincy Adams because he didn’t like Monroe. I don’t know how many people voted for that elector, if Massachusetts even elected their electors by popular vote.
The absolutely pedantic final answer is either DeWitt Clinton or John Quincy Adams in 1820. Which one counts the “losing candidate?” You pick.
I’m assuming this line was added because of the 1912 election?
In that year, William Taft became the only incumbent President to come in third place in his re-election attempt (with 23.2% of the popular vote and only eight electoral votes).